Connect to RDS through VPC Peering

Question

I'm attempting to establish a VPC Peering connection from my local computer to an RDS Instance. I have two VPCs in my AWS account, VPC1 and VPC2, however only VPC1 is connected to my local network via DirectConnect. All of my infrastructure is located in VPC2, and according to the plan, I must use VPC1 to access to it from my local machine.

I've set up a route in the peering connection to forward IP-based requests for a specific address range to VPC2. However, since I just know the endpoint and not the IP Address for RDS, this doesn't really help me. Although I haven't found any material that explains how to resolve this issue, I am speculating that some combination of DNS, Routing, Networking, and Peering will be able to resolve this issue.

Has this problem ever been resolved before? Is there any documentation out there that outlines what has to be done?

Update: The precise issue is that I am unable to connect from my local machine to the RDS instance. The SQL Client I'm using can't connect with a timeout problem, for instance, if I use the RDS Endpoint as the server for my connection. I have a hunch that traffic to VPC2 is not being routed properly, but I have no idea how to prove it.

I don't know how OnPrem is set up with regard to DNS, but I have 4 hosted zones in Route53 with a variety of URLs. On my local network, I may resolve anything that I configured in Route53 by host name.

Likewise, I'm not sure how DirectConnect's network configuration has been set up (full VPN tunnel or otherwise).

However, that stuff functions as far as DNS and the network connections across AWS are concerned. I can successfully resolve infrastructure in VPC1; but, it appears that I am unable to get traffic to traverse the peering connection in the manner I would anticipate.